detect a mean change if one occurs. Three different approaches are used to solve this problem : OLS, GLS and a frequency domain method.

Transcription

1 Abstract HUH, SEUNGHO. Sample Size Determination and Stationarity Testing in the Presence of Trend Breaks. (Under the direction of Professor David A. Dickey.) Traditionally it is believed that most macroeconomic time series represent stationary fluctuations around a deterministic trend. However, simple applications of the Dickey-Fuller test have, in many cases, been unable to show that major macroeconomic variables are stationary univariate time series structure. One possible reason for non-rejection of unit roots is that the simple mean or linear trend function used by the tests are not sufficient to describe the deterministic part of the series. To address this possibility, unit root tests in the presence of trend breaks have been studied by several researchers. In our work, we deal with some issues associated with unit root testing in time series with a trend break. The performance of various unit root test statistics is compared with respect to the break induced size distortion problem. We examine the effectiveness of tests based on symmetric estimators as compared to those based on the least squares estimator. In particular, we show that tests based on the weighted symmetric estimator not only eliminate the spurious rejection problem but also have reasonably good power properties when modified to allow for a break. We suggest alternative test statistics for testing the unit root null hypothesis in the presence of a trend break. Our new test procedure, which we call the bisection method, is based on the idea of subgrouping. This is simpler than other methods since the necessity of searching for the break is avoided. Using stream flow data from the US Geological Survey, we perform a temporal analysis of some hydrologic variables. We first show that the time series for the target variables are stationary, then focus on finding the sample size necessary to

2 detect a mean change if one occurs. Three different approaches are used to solve this problem : OLS, GLS and a frequency domain method. A cluster analysis of stations is also performed using these sample sizes as data. We investigate whether available geographic variables can be used to predict cluster membership.

3

4 To my wife, son, parents and parents-in-law ii

5 iii Biography Seungho Huh was born in Seoul, Korea in March 21, In 1987, he graduated from Seoul National University with a Bachelor s degree in Statistics. After receiving his Master s degree in Statistics at Seoul National University in 1989, he started to work for Korea Telecom as a member of the technical staff. He married Seungshin in 1993 and had a son, Jungwoo, in Since the fall semester of 1996, he has been studying for the Ph.D. degree at North Carolina State University, under the direction of Dr. David A. Dickey.

6 iv Acknowledgements I would be grateful to my advisor Dr. Dickey from the bottom of my heart. My research became fruitful thanks to his helpful instruction and advice. Without meeting him, I must have been frustrated in writing this dissertation. He also showed me what a scholar should look like. I was always impressed with his enthusiasm for both teaching and research. I would like to thank my committee members, Dr. Bhattacharyya, Dr. Gumpertz and Dr. Pantula, for their thoughtful comments and suggestions that greatly improved the quality of this dissertation. Another memorable thing during my stay in the department was Dr. Pantula s devotional concern for students as the Director of Graduate Program. My wife, Seungshin, has always been encouraging and supporting me during the whole period of study. If it had not been for her, my hard days of schoolwork would have been worse. Always on my side was Seungshin and my son Jungwoo when I was worn out. I am sure they would be happier than me with obtaining my Ph.D. degree. Lastly, but not least importantly, this honor is surely attributed to my parents and parents-in-law who brought up me and Seungshin with their whole hearts. Throughtout the years of my studying abroad, they affectionately supported me from a distance. They deserve to be congratulated on their son or son-in-law s achievement.

7 v Contents List of Tables List of Figures vii ix 1 Introduction Literaturereview Outlineofresearch Comparison of Break Induced Size Distortions among Unit Root Test Statistics Introduction Datawithabreakinlevel Usingmeanadjustedstatistics Effectofignoringthebreak Analysisofexpectations Usinglineartrendadjustedstatistics Using lagged first differences Datawithabreakintrend Empiricalpowersallowingforabreak Conclusion Alternative Method for Testing the Unit Root Null Hypothesis in the Presence of a Break Introduction Subgrouping of data Bisectionmethod Empiricalsizeandpowerresults Datawithabreakinlevel Meanadjustedcase Lineartrendadjustedcase Datawithabreakinslope... 43

8 vi 3.5 Empiricalapplications Summary Temporal Analysis of Hydrologic Variability in Stream Flow Data Introduction Preliminaryresults Descriptionofdata Modelfitting Samplesizefordetectingthelevelshift MethodI:Ordinaryleastsquares MethodII:Generalizedleastsquares Noncentralityparameter Comparison of theoretical and empirical powers MethodIII:Frequencydomainmethod Resultsforstreamflowdata Samplesizeestimates Measureofaccuracy Clusteranalysis Summary Conclusion 124 References 126 Appendices 130

9 vii List of Tables 2.1 EmpiricalsizesforModelIusingmeanadjustedstatistics Empirical sizes for Model I using linear trend adjusted statistics Empirical sizes for Model I using lagged first differences (Weighted Symmetric estimator) Empirical sizes for Model II using linear trend adjusted statistics EmpiricalpowersforModelIusingthemodifiedtest Empirical size and power for DGP (3.1) using various subgroups (OLS, n = 100, c =36) Empirical size and power for DGP (3.1) using various subgroups (SS, n = 100, c =37) Empirical size and power for DGP (3.1) using various subgroups (WS, n = 100, c =37) Empirical size and power for DGP (3.1) using τ w and τw Empirical size and power for DGP (3.1) using various statistics (n = 100, c =50) Empirical size and power for DGP (3.1) using τ w,τ and τw,τ Empirical size and power for DGP (3.2) using τ w,τ and τw,τ Empirical size and power for DGP (3.2) using various statistics (n = 100, c =50) Testresultsfortheunitrootnullhypothesis Descriptiveinformationforthestationsofinterest Seriessuggestinghigherordermodeland/orlineartrend Estimates of the lag 1 autoregressive coefficient ρ for each station and targetvariable Sample sizes to detect a level shift of size δ = kσ for various values of β and ρ (MethodI) Theoretical and empirical powers for some values of ρ, n and k =1 usingmethodii Sample sizes for detecting the level shift (k =1,β=.2, Method I).. 105

10 4.7 Sample sizes for detecting the level shift (k = 1, β =.2, Method II) Sample sizes for detecting the level shift (δ = 3, β =.2, Method III) Sample sizes for detecting the level shift (k =1,β=.2, Method I using t distribution) Correlation matrix for the 4 variables n 1, n 2, n 3 and n 4 in Table Correlation matrix for the 4 variables n 1, n 2, n 3 and n 4 in Table 4.6 (outlyingstationsexcluded) Principal component vectors for the 4 variables n 1, n 2, n 3 and n 4 in Table4.6(outlyingstationsexcluded) Clustering of the stations by the descriptive statistics (complete linkage)112 viii

11 ix List of Figures 2.1 Random walk with a break in level (Model I, c =50) Empirical sizes for Model I with m = 10 using mean adjusted statistics Expectations of quadratic forms, their ratios and pivotal statistics (n = 100) Empirical sizes for Model I with m = 10 using linear trend adjusted statistics Empirical sizes for Model I using lagged first differences (Weighted Symmetric estimator) Random walk with a break in drift (Model II, c =50) Empirical sizes for Model II with m = 2 using linear trend adjusted statistics Empirical size and power for DGP (3.1) using various subgroups (OLS, n = 100, c =36) Empirical size and power for DGP (3.1) using various subgroups (SS, n = 100, c =37) Empirical size and power for DGP (3.1) using various subgroups (WS, n = 100, c =37) Empirical distributions of test statistics (WS, n = 100, c = 75, θ = 5 and ρ =.7; T for τ w, M for τw, L for τ w,1 and R for τ w,2 ;MoverlaysL.) Empirical size and power of test statistics (WS, n = 100, c =75and θ= 5; T for τ w, M for τw, L for τ w,1 and R for τ w,2 ) Data from a break-in-level model (DGP (3.1), ρ = 0.5, c = 50) Data from a break-in-level model (DGP (3.1), ρ = 1, c = 50) Data from a break-in-slope model (DGP (3.2), ρ = 0.5, c = 50) Data from a break-in-slope model (DGP (3.2), ρ = 1, c = 50) Empirical size and power for DGP (3.1) using τ w (1 group) and τw (2 groups) (θ =2.5, λ = c/n) Empirical size and power for DGP (3.1) using τ w (1 group) and τw (2 groups) (θ =5, λ =c/n)... 68

12 x 3.12 Empirical size and power for DGP (3.1) using τ w (1 group) and τ w (2 groups) (θ = 10, λ = c/n) Empirical size and power for DGP (3.1) using τ w,τ (1 group) and τ w,τ (2 groups) (θ =2.5, λ = c/n) Empirical size and power for DGP (3.1) using τ w,τ (1 group) and τ w,τ (2 groups) (θ =5,λ=c/n) Empirical size and power for DGP (3.1) using τ w,τ (1 group) and τ w,τ (2 groups) (θ = 10, λ = c/n) Empirical size and power for DGP (3.2) using τ w,τ (1 group) and τ w,τ (2 groups) (γ =0.5, λ = c/n) Empirical size and power for DGP (3.2) using τ w,τ (1 group) and τ w,τ (2 groups) (γ =1,λ=c/n) Empirical size and power for DGP (3.2) using τ w,τ (1 group) and τ w,τ (2 groups) (γ =2,λ=c/n) Time series data for station Distributionofthedescriptivestatisticsacrossthestations Theoretical powers for some values of ρ (Method II) Comparison of sample sizes between TDM and FDM (triangular weights oflength15;serialnumbersintable4.1usedforsites) Comparison of sample sizes between TDM and FDM (flat weights of length25;serialnumbersintable4.1usedforsites) The first two principal components for the 4 variables n 1, n 2, n 3 and n 4 intable4.6(outlyingstationsexcluded) Clustering of the stations by the descriptive statistics (complete linkage) Clustering of the stations by the sample sizes (single linkage; maximizing CCC) Clustering of the stations by the sample sizes (single linkage; The largestclustercontainsnomorethan25stations.) Scatter plots of elevation, drainage area and sample size from Method I(k=1) Scatter plots of elevation, drainage area and sample size from Method I(δ=3)

13 Chapter 1 Introduction 1.1 Literature review In the field of time series analysis, there are many good references and textbooks available among which are Priestley (1981), Wei (1990), Box, Jenkins and Reinsel (1994), Hamilton (1994) and Fuller (1996). Of those mentioned, only Box, Jenkins and Reinsel (1994), Hamilton (1994) and Fuller (1996) address the topic of unit roots. The analysis of time series with a unit root became popular following, among others, the work of Dickey (1976). He shows that under the null hypothesis H 0 : ρ =1 in the model Y t = ρy t 1 + e t (Y 0 =0,e t NI(0,σ 2 )), n(ˆρ 1) d 0.5(T 2 1)/G where ˆρ is the least squares estimator of ρ, T = i=1 2γi U i, G = i=1 γ 2 i U 2 i, γ i =2( 1) i+1 /π(2i 1) and U i NI(0, 1). Dickey (1976) finds the distribution of n(ˆρ µ 1) and n(ˆρ τ 1) where ˆρ µ and ˆρ τ are the least squares estimators of ρ when the fitted model contains a nonzero mean term and a linear trend term respectively. He also tabulates the distribution of t type statistics under the above 3 models. Critical values of these distribuions can be found in Dickey (1976) and Fuller (1996). See Dickey and Fuller (1979) for more information. The random variables T = W (1) and G = 1 0 W 2 (t)dt can be expressed as functionals of a standard Wiener process W (t). Said and Dickey (1984) consider testing for unit roots in the general ARMA(p, q)

14 2 model of unknown orders p and q. They show that fitting a high order autoregressive model is an appropriate way to test for a unit root in a model of unknown order. Their method is commonly known as the augmented Dickey-Fuller (ADF) test although ADF is also mentioned in Dickey and Fuller (1979) for pure autoregressions.. Gonzalez-Farias (1992) considers maximum likelihood estimation of the parameters in autoregressive time series. Her paper studies the maximizers of the exact stationary likelihood function when the input data are from a unit root process. She proposes a new unit root test based on these estimators and derives its limiting distribution. Dickey, Hasza and Fuller (1984) discuss the properties of another type of estimator known as the simple symmetric estimator. Park and Fuller (1995) study the weighted symmetric estimator which is another in the class of symmetric estimators. Pantula, Gonzalez-Farias and Fuller (1994) compare the performance of these and other unit root test criteria and determine that the weighted symmetric estimator and the unconditional maximum likelihood estimator provide the most powerful tests against the stationary alternative. Nelson and Plosser (1982) investigate a collection of important macroeconomic time series asking if they are better characterized as stationary fluctuations around a deterministic trend or as nonstationary processes that have no tendency to return to a deterministic path. They apply the augmented Dickey-Fuller test to 14 major macroeconomic time series. Their study, which finds that most macroeconomic variables have a univariate time series structure with a unit root, is followed by a series of empirical analyses with similar findings. Several researchers have tried to explain these rather interesting results, believing that most macroeconomic time series represent stationary fluctuations around a deterministic trend. Two possible reasons for non-rejection of unit roots are that (1) there are not enough observations to endow the tests with sufficient power and (2) the simple mean or linear trend function used by the tests are not sufficient to describe the deterministic part of the series. To address the second possibility, the topics of unit

15 3 root testing and trend breaks are combined by Perron (1989). In the pioneering work of Perron (1989), he considers the null hypothesis that a time series has a unit root with possibly nonzero drift against the alternative that the process is trend stationary. Allowing, under both the null and alternative hypotheses, for the presence of a onetime change in the trend function, Perron (1989) derives test statistics which can distinguish the two hypotheses when such a break is present. He applies these tests to the Nelson and Plosser (1982) data set and rejects the unit root hypothesis for 11 out of the 14 series. He argues that the failure of the usual Dickey-Fuller test to reject the unit root null hypothesis reflects not an actual unit root, but instead that the data are trend stationary around a broken trend. This has come to be called the Perron phenomenon. Since Perron assumes the break point is known a priori and treated as exogenous, criticism has emerged from some authors such as Banerjee, Lumsdaine and Stock (1992), Christiano (1992) and Zivot and Andrews (1992). The most notable of them is Christiano (1992) who argues that the choice of break dates has to be viewed as being correlated with the data. He presents some algorithms for selecting the break date endogenously. A bootstrap approach that takes into account pretest data examination is used to test the null hypothesis of no trend break. Banerjee, Lumsdaine and Stock (1992) also treat the break date as unknown a priori and suggest recursive and sequential tests of the unit root null hypothesis. Their theoretical results are similar to those of Zivot and Andrews (1992). Zivot and Andrews (1992) transform Perron s (1989) unit root test, which is conditional on structural change at a known point in time, into an unconditional unit root test in which the break point is estimated rather than fixed. They check all possible break points and take the minimum t statistic. The data series considered by Perron (1989) are reanalyzed using their estimated break point test statistic. Perron (1994) is a good summarizing reference for the issue of unit roots and

16 4 structural change. He presents works by himself and many other researchers. Perron s (1997) work is closely related to that of Banerjee et al (1992) and that of Zivot and Andrews (1992) in that similar procedures and series are analyzed. He first reexamines his findings from Perron (1989). Unlike his previous study, Perron (1997) assumes that the date of a possible break is not fixed a priori but instead is unknown. He considers various methods to select the break point and the truncation lag parameter. Most of the rejections reported in Perron (1989) are confirmed using his new approach. Additional tests for a unit root allowing for a break at an unknown time are given by Vogelsang and Perron (1998). They focus on the additive outlier approach where the break is sudden, as opposed to the innovational outlier approach where the change occurs slowly over time. Recently, Leybourne, Mills and Newbold (1998) discover and analyze what they call the converse Perron phenomenon. They show that routine application of the Dickey-Fuller test can lead to a severe problem of spurious rejection of the null hypothesis when the data are generated from a random walk with a trend-break near the beginning. Leybourne et al (1998) also show that this is not an end-effects problem in the ordinary least squares estimation since their results hold asymptotically for the break size growing at an appropriate rate. For a structural break of fixed size, on the other hand, Amsler and Lee (1995) suggest that the asymptotic distributions of the usual Dickey-Fuller tests under ρ = 1 are unaffected and so the spurious rejection problem exists only in finite samples. Chapter 2 of this dissertation is closely related to Leybourne et al (1998) in that we compare the performance of the symmetric estimator with that of the ordinary least squares estimator in the problem of spurious rejection.

17 5 1.2 Outline of research In our work, we concentrate on analyzing time series data with a trend-break. Three somewhat independent papers, Chapters 2, 3 and 4, are related by their focus on the trend-break issue. In Chapter 2, we compare the performance of various unit root test statistics in the break induced size distortion problem. Leybourne et al (1998) show that, if the data generating process is a random walk with a break near the beginning, standard Dickey-Fuller (1979) tests based on the least squares estimator have serious size distortion. In Chapter 2, we examine the performance of alternative tests based on symmetric estimators. In particular we show that tests based on the weighted symmetric estimator not only eliminate spurious rejection in these problem data sets but also have reasonably good power properties when modified to allow for a break. In Chapter 3, we suggest alternative test statistics for testing the unit root null hypothesis in the presence of a trend-break. Our new test procedure which we call the bisection method is based on the idea of subgrouping. The idea here is to split the data in half and look at the minimum of the resulting two unit root test statistics. This avoids the necessity of searching for the break. It uses all the data in the sense that the minimum is chosen, but clearly is not efficient in its use of the data. We anticipate paying a price in power for a gain in simplicity. Considering some data generating processes, we display empirical size and power results from simulation. We also apply our bisection method to the well-known Nelson and Plosser (1982) data set and compare the results with those of others. The simple bisection method rejects unit roots in several, but not all, of the series for which the more complicated search methods reject. In Chapter 4, we analyze stream flow data from the US Geological Survey. We perform a temporal analysis of some hydrologic variables of interest to USGS scientists. We are particularly interested in the required sample size to detect a break, or shift, of the mean in the presence of autocorrelation. We apply three alternative

18 6 approaches to obtain the sample size. The first approach is a simple method using ordinary least squares theory that assumes a known error variance. The second approach uses generalized least squares and is more realistic in that it treats the error variance as unknown. The third approach is a frequency domain method for which identification of the autocorrelation structure is not needed. Chapter 4 finishes with a cluster analysis, performed to obtain clusters based on the sample sizes. The cluster pattern does not suggest any particular physiographic segregation. This is consistent with a regression analysis which shows no influence of available physiographic variables on the sample sizes.

19 Chapter 2 Comparison of Break Induced Size Distortions among Unit Root Test Statistics 2.1 Introduction Recently there has been much interest in testing for a unit root in a time series that has a trend break. Leybourne, Mills and Newbold (1998) study the behavior of standard unit root tests when the data generating process contains a trend break not accounted for by the fitted model. For data consisting of a random walk with a shift in level, they report empirical sizes less than the nominal level when the shift is not too near the beginning of the series. However, if the shift is near the beginning of the series, they report too many rejections of the unit root null hypothesis. Thus a unit root process with an early level shift is too often declared stationary. Leybourne et al (1998) call this the converse Perron phenomenon in contrast to the well known Perron phenomenon investigated by Perron (1989). If we switch from tests based on the least squares estimator to more powerful tests based on the simple symmetric or weighted symmetric estimators (see Pantula,

20 8 Gonzalez-Farias and Fuller (1994)), we find that this phenomenon decreases dramatically, seeming to disappear in the weighted symmetric case. In section 2.2, we examine the simple random walk model with a break in level. We consider the simple symmetric estimator ρ s and the weighted symmetric estimator ρ w of the lag 1 autoregressive coefficient ρ. Empirical sizes for the pivotal statistics τ s and τ w associated with ρ s and ρ w, respectively, are presented to compare with those for τ µ, the ordinary least squares test statistic. To gain some insight into the empirical results, we examine the expected values of the quadratic forms constituting the test statistics. This analysis includes τ s,τ and τ w,τ, the linear trend adjusted versions of τ s and τ w, respectively, to compare with τ τ, the linear trend adjusted version of τ µ. An augmented test with lagged first differences is introduced to extend the results to more general cases. In section 2.3, we consider the random walk model with a break in trend, using τ s,τ and τ w,τ as test statistics to obtain empirical sizes. The results are compared with those for τ τ. Section 2.4 presents some power results for a test based on the weighted symmetric estimator, modified to allow for a break. Finally we make some concluding remarks in section Data with a break in level In this section, following Leybourne et al (1998), we consider a data generating process (DGP) Model I : Y t = mσi(t >c)+x t, X t = X t 1 +e t, t =1,2,..., n where the e t are normal independent (0,σ 2 ) random variables, m is the size of the break as a multiple of σ and c = nλ isthetimeofthebreak. Theestimatorsofthe lag 1 autoregressive coefficient ρ are invariant to σ so we can assume σ =1. This model corresponds to Model (A) with µ = 0 under the null hypothesis of Perron (1989). Figure 2.1 presents some typical time series data generated from Model I.

21 Using mean adjusted statistics As in Leybourne et al (1998), to test the unit root null hypothesis, we consider ˆρ µ = nt=2 (Y t Ȳ )(Y t 1 Ȳ ) nt=2 (Y t 1 Ȳ )2 (2.1) which is an ordinary least squares (OLS) estimator of ρ in the non-zero mean first order autoregressive process Y t = µ + ρy t 1 + e t. (2.2) We use, as a test statistic, the associated pivotal statistic τ µ = ˆρ µ 1 s.e. = ˆρ µ 1 [ n t=2 (Y t 1 Ȳ )2 ] 1 s 2 (2.3) where s 2 = 1 n [Y t n 3 Ȳ ˆρ µ(y t 1 Ȳ )]2. t=2 Using the test statistic τ µ, Leybourne et al (1998) reported the empirical sizes of nominal 5% level tests for various values of break size, m, and break time, c. They found that, as m increases, an increasingly severe phenomenon of spurious rejection of the unit root null hypothesis emerges. They also indicated that the earlier is the break, the greater is the rejection rate. Our empirical sizes for τ µ are in close agreement with those reported by Leybourne et al (1998). We consider alternative symmetric estimators ρ s and ρ w (Fuller, 1996) of ρ in (2.2) and their associated pivotal statistics τ s and τ w. As will be seen, their performance is often quite superior to that of τ µ. For the process (2.2), the simple symmetric (SS) estimator can be written as ρ s = nt=2 y t y t 1 n 1 t=2 yt (y2 1 + yn 2) This is from the least squares regression of Y t Ȳ on Y t 1 Ȳ. The regression of Y t on 1, Y t 1 gives a slightly different estimator, differing trivially from (2.1).

22 10 where y t = Y t Ȳ. The pivotal statistic for the SS estimator is τ s = ρ s 1 s.e. = ρ s 1 σ 2 s [ (2.4) n 1 t=2 y2 t (y2 1 + yn 2)] 1 where σ 2 s = = 1 n n 2 [ n 1 1 t=2 2 (y t ρ s y t 1 ) t=1 2 (y t ρ s y t+1 ) 2 ] 1 n n 2 [ (y t ρ s y t 1 ) (1 ρ2 s)(y1 2 yn)]. 2 t=2 The weighted symmetric (WS) estimator for the process (2.2) is nt=2 y t y t 1 ρ w = n 1 t=2 y2 t + 1 nt=1 y 2 n t and the associated pivotal statistic is (2.5) τ w = ρ w 1 s.e. = ρ w 1 σ 2 w ( n 1 t=2 y2 t + 1 (2.6) nt=1 y 2 n t ) 1 where and w t = t 1 n estimators. τ w. σ w 2 = 1 n n 2 [ w t (y t ρ w y t 1 ) 2 + (1 w t+1 )(y t ρ w y t+1 ) 2 ] = t=2 n 1 t=1 1 n 2 [ n t=2(y t ρ w y t 1 ) 2 +(1 ρ 2 w)(y n n yt 2 )]. t=1 for t =1,,n. See Fuller (1996) for detailed discussion of symmetric In Table 2.1, we show empirical sizes of nominal 5% level tests using τ µ,τ s and We consider the same values of m and c as those used by Leybourne et al (1998). Simulations are based on 5,000 replications at each (m, c) combination so the standard error of an empirical size is less than.25/5000 =.007. The sample size per replication is n = 100. Critical values used are those for the mean removed case in Fuller (1996). Data are generated with X 0 =0sothatX 1 =e 1 N(0, 1). Our τ µ simulations match those of Leybourne et al (1998), however we find that the spurious rejection problem is far less severe in the symmetric estimators. For

23 11 τ s, the proportion of rejections shows symmetry around c = 50 and gets larger as m increases. The rejection rate is higher for c values in the first or last part than for those in the middle of the data. The τ s rejection rate is much smaller than that of τ µ for early breaks but larger for late breaks. For τ w, the rejection rate is almost invariant to the break size m and shows little size distortion. Most importantly, compared to τ µ, τ w gives satisfactory performance in that the largest empirical size is around (m=5) and most of the others are less than the nominal level Figure 2.2 shows empirical sizes of the three tests using mean adjusted statistics for model I with m = Effect of ignoring the break To obtain some insight into the empirical results, we examine some statistical properties of the test statistics when the break is ignored. As suggested by Amsler and Lee (1995), the asymptotic distributions of the usual Dickey-Fuller tests under ρ = 1 are unaffected by a structural break of fixed size. We now show that this also holds good for the tests based on the symmetric estimators. Thus the spurious rejection problem exists only in finite samples and the effect of ignoring the break may differ between the OLS estimator and the symmetric estimators. Let x t = X t X in Model I where X = 1 nt=1 X n t. Then we know that Ȳ = X +(1 λ)m and y t = x t (1 λ)m x t + λm if t c = nλ if t>c=nλ. Recall that the OLS estimator of ρ is ˆρ µ = nt=2 y t y t 1 nt=2 y 2 t 1

24 12 and ˆρ µ 1= nt=2 y t 1 (y t y t 1 ) nt=2 y 2 t 1 N µ D µ (2.7) By some straightforward algebra, the numerator of ˆρ µ 1 can be written as n y t 1 (y t y t 1 ) = t=2 n t=2 nλ+1 x t 1 (x t x t 1 ) (1 λ)m e t + λm +mx nλ (1 λ)m 2. t=2 n e t t=nλ+2 Therefore 1 n y t 1 (y t y t 1 ) = 1 n x t 1 (x t x t 1 )+O p ( 1 ). n t=2 n t=2 n The denominator of ˆρ µ 1 can be written as n yt 1 2 = t=2 n x 2 nλ+1 t 1 2m n x t 1 +2λm x t 1 t=2 t=2 t=2 +(1 λ) 2 m 2 nλ + λ 2 m 2 (n nλ 1). Using the fact that and we have nλ+1 1 x n 2 t 1 = O p ( 1 ) n 1 n 2 t=2 n 1 x t 1 = O p ( t=2 n n ), 1 n y n 2 t 1 2 = 1 n x 2 t=2 n 2 t 1 + O p ( 1 ). t=2 n (2.8) Combining the results, n(ˆρ µ 1) = 1 nt=2 y n t 1 (y t y t 1 ) 1 (2.9) = = 1 n 1 n nt=2 y 2 n 2 t 1 nt=2 x t 1 (x t x t 1 )+O p ( 1 n ) 1 nt=2 x 2 n 2 t 1 + O p ( 1 n ) nt=2 x t 1 (x t x t 1 ) 1 n 2 nt=2 x 2 t 1 n(ˆρ µ,0 1) + O p ( 1 n ) + O p ( 1 n )

25 13 where ˆρ µ,0 stands for the OLS estimator of ρ with no break. Recall that the SS estimator of ρ is nt=2 y t y t 1 ρ s = n 1 t=2 yt (y2 1 + yn 2) and nt=2 y t y t 1 n 1 t=2 ρ s 1 = y2 t 1 2 (y2 1 + y2 n ) n 1 t=2 y2 t (y2 1 + yn 2) = 1 nt=2 (y 2 t y t 1 ) 2 n 1 t=2 yt (y2 1 + yn 2) N s (2.10) D s by rearranging the terms in the numerator. The numerator of ρ s 1 can be rewritten as 1 n t y t 1 ) 2 t=2(y 2 = 1 n (x t x t 1 ) 2 m(x nλ+1 x nλ ) m2 2 t=2 2. Therefore 1 n n { 1 (y t y t 1 ) 2 } = 1 n (x t x t 1 ) 2 m 2 t=2 2n t=2 n e nλ+1 m2 2n = 1 n n { 1 (x t x t 1 ) 2 } + O p ( 1 ). (2.11) 2 t=2 n We notice that the effect of ignoring the break does not depend on the break time c = nλ here. Since y 2 1 = x2 1 2(1 λ)mx 1 +(1 λ) 2 m 2 and y 2 n = x 2 n +2λmx n + λ 2 m 2, we can easily show that 1 n 2 y2 1 = 1 1 n 2 x2 1 + O p ( n n ) and 1 n 2y2 n = 1 1 n 2x2 n +O p ( n ). (2.12) n By (2.8) and (2.12), we have n 1 1 n { 2 t=2 yt (y2 1 + y2 n )} = 1 n 1 n { 2 t=2 x 2 t (x2 1 + x2 n )} + O p( 1 n ) (2.13)

26 14 Combining (2.11) and (2.13), n( ρ s 1) = = = 1 nt=2 (y 2n t y t 1 ) 2 1 { n 1 n 2 t=2 yt (y2 1 + yn)} 2 nt=2 (x t x t 1 ) 2 + O p ( 1 ) n t=2 x 2 t (x2 1 + x 2 n)} + O p ( 1 n ) nt=2 (x t x t 1 ) 2 t=2 x2 t (x2 1 + x 2 n )} + O p( 1 ) n 1 2n 1 { n 1 n 2 1 2n 1 { n 1 n 2 n( ρ s,0 1) + O p ( 1 n ) (2.14) where ρ s,0 stands for the SS estimator of ρ with no break. and As to the WS estimator of ρ, we recall that ρ w 1= ρ w = nt=2 y t y t 1 n 1 t=2 yt nt=1 y 2 n t nt=2 y t y t 1 n 1 t=2 y 2 t 1 n nt=1 y 2 t n 1 t=2 y2 t + 1 n nt=1 y 2 t N w D w. (2.15) By some straightforward algebra, the numerator of ρ w 1 can be written as n n 1 y t y t 1 yt 2 1 n yt 2 = 1 n t y t 1 ) t=2 t=2 n t=1 2 t=2(y (y2 1 + yn) 2 1 n yt 2. n t=1 By (2.12), we can show that 1 n y2 1 = 1 n x2 1 + O p( 1 ) n and 1 n y2 n = 1 n x2 n +O p( 1 n ). (2.16) Therefore 1 n n ( y t y t 1 t=2 n 1 t=2 by (2.8), (2.11) and (2.16). On the other hand, yt 2 1 n yt 2 n ) = 1 n t=1 n { 1 t x t 1 ) 2 t=2(x (x2 1 + x2 n ) 1 n x 2 t n } t=1 +O p ( 1 ) (2.17) n n 1 1 n ( y 2 2 t + 1 n t=2 n t=1 yt 2 )= 1 n 1 n 2( t=2 x 2 t + 1 n n t=1 x 2 t )+O p( 1 n ) (2.18)

27 15 by (2.8). Combining (2.17) and (2.18), n( ρ w 1) = = 1 n ( n t=2 y t y t 1 n 1 t=2 y2 t 1 nt=1 y 2 n t ) 1 ( n 1 n 2 t=2 y2 t + 1 nt=1 y 2 n t ) 1 { 1 nt=2 (x n 2 t x t 1 ) (x2 1 + x2 n ) 1 n 1 ( n 1 n 2 t=2 x2 t + 1 nt=1 x 2 n t )+O p ( 1 1 n = ( n t=2 x t x t 1 n 1 1 ( n 1 n 2 t=2 x 2 t + 1 n 1 n = ( n t=2 x t x t 1 n 1 1 ( n 1 n 2 n( ρ w,0 1) + O p ( 1 ) n t=2 x2 t 1 n nt=1 x 2 t } + O p( 1 n ) n ) nt=1 x 2 t )+O p( 1 n ) nt=1 x 2 t )+O p ( 1 t=2 x2 t 1 nt=1 x 2 n t ) t=2 x2 t + 1 nt=1 x 2 n t ) where ρ w,0 stands for the WS estimator of ρ with no break. n ) + O p ( 1 n ) (2.19) Thus the effect of an added break is O p (1/ n) in all cases with the numerator of n( ρ s 1) having an even faster convergence rate, O p (1/n). The studentized test statistics can be written as τ µ = ˆρ µ 1 s.e. = n(ˆρ µ 1) 1 n 2 nt=2 y 2 t 1 s 2 = {n(ˆρ µ,0 1) + O p ( 1 n )} 1 nt=2 x 2 n 2 t 1 + O p ( 1 s 2 τ µ,0 + O p ( 1 n ), n ) (2.20) τ s = ρ s 1 s.e. = n( ρ s 1) 1 { n 1 n 2 t=2 yt (y2 1 + yn)} 2 (2.21) σ 2 s = {n( ρ s,0 1) + O p ( 1 n )} 1 { n 1 n 2 t=2 x2 t (x2 1 + x 2 n )} + O p( 1 σ 2 s τ s,0 + O p ( 1 n ) n ) and τ w = ρ w 1 s.e. = n( ρ w 1) 1 ( n 1 n 2 t=2 yt nt=1 y 2 n t ) (2.22) σ 2 w

28 16 = {n( ρ w,0 1) + O p ( 1 n )} 1 ( n 1 n 2 t=2 x 2 t + 1 n σ 2 w τ w,0 + O p ( 1 n ) nt=1 x 2 t )+O p ( 1 n ) where τ µ,0, τ s,0 and τ w,0 represent the test statistics with no break. We use the fact that s 2, σ s 2and σ2 w are consistent regardless of the break under ρ =1. Although the effect of ignoring the break in the numerator of n( ρ s 1) is O p ( 1 ), n the effect is O p ( 1 n ) in all of the studentized test statistics. The spurious rejection problem disappears at the same rate for all tests, as n becomes larger Analysis of expectations We next analyze the expectations of the quadratic forms constituting the test statistics to gain some more clues with respect to the empirical results. These are only clues, as the power also depends on the shapes of the distributions tails. Define ˆρ µ 1 N µ /D µ, ρ s 1 N s /D s and ρ w 1 N w /D w. Some straightforward but tedious algebra gives E[N µ ] = nσ 2 [ n m2 (1 λ) ] n E[D µ ] = n 2 σ 2 [ n + 1 3n 1 2 6n + m2 λ(1 λ) m2 λ 2 ] 3 n n 2 E[N s ] = nσ 2 ( n m2 2n ) E[D s ] = n 2 σ 2 [ n + 1 3n 1 2 6n + m2 λ(1 λ) m2 λ 2 + m 2 (1 λ) 2 ] 3 n 2n 2 E[N w ] = nσ 2 [ n 2m2 λ(1 λ) ] 2 n E[D w ] = n 2 σ 2 [( n + 5 6n 1 +1 )(n 2 3n3 n )+ 2 3n 1 2 n n 4 + m2 λ(1 λ)(n +1) m 2 λ 2 m 2 (1 λ) 2 ] n 2 where λ = c/n is the proportion of observations before the break. Figure 2.3 presents the expectations of the quadratic forms, their ratios and pivotal statistics constructed by replacing quadratic forms with their expectations.

29 17 The expectations of the denominators above are concave functions of λ and symmetric around λ =0.5. They are very similar as we can see in the second column of Figure 2.3. This is expected because the denominators can be written as D µ = D s = D w = n 1 yt 2 + y2 1 t=2 n 1 yt t=2 2 (y2 1 + yn) 2 n 1 yt n yt 2 t=2 n t=1 The major differences are in the numerators. E[N µ ] is an increasing function of λ and E[N s ] does not depend on λ. E[N w ] is a convex function of λ and symmetric around λ = 0.5. Notice that lower values of the numerators suggest higher rates of rejecting H 0 : ρ = 1 as we are using left tailed tests. If we consider the ratio of the expectations of the numerators and the denominators, E[N µ ]/E[D µ ] is a concave function of λ which is negative and not symmetric around λ =0.5. Its absolute value is larger for λ near 0 than for λ near 1. As λ increases from 0 to 1, its absolute value decreases at first and then increases. This corresponds to the empirical results of high τ µ rejection rates for early breaks. E[N s ]/E[D s ] is a symmetric concave function of λ. Its absolute values are larger for λ near 0 or 1 than for λ near 0.5. E[N w ]/E[D w ] is also a symmetric concave function of λ, becoming nearly constant as n increases. The absolute values of E[N s ]/E[D s ] are much larger than those of E[N w ]/E[D w ] for λ near 0 or 1. This also corresponds to the empirical results that τ s and τ w show symmetry around λ =0.5withτ s giving higher rejection rates than τ w for λ near 0 or 1. All three standard errors in (2.3), (2.4) and (5.1) are of the form MSE s.e. = D where MSE stands for the mean square error and D denotes D µ, D s or D w. Recalling that the mean square error is a consistent estimator of σ 2, we define an approximate

30 18 standard error, a.s.e., as a.s.e. σ2 E[D]. We find that a.s.e. is a symmetric convex function of λ and that (E[N µ ]/E[D µ ])/a.s.e. and (E[N s ]/E[D s ])/a.s.e. show similar shapes to E[N µ ]/E[D µ ]ande[n s ]/E[D s ]respectively. On the other hand, (E[N w ]/E[D w ])/a.s.e. has a little different shape from that of E[N w ]/E[D w ]. The former is convex whereas the latter is nearly flat but concave. These graphs are displayed as pivotal statistics in the right column of Figure Using linear trend adjusted statistics Leybourne et al (1998) also considered ˆρ τ which is the OLS estimator of ρ in the model Y t = µ + βt + ρy t 1 + e t. (2.23) Alternatively the first order autoregressive model around a linear trend can be written as Y t µ βt = ρ[y t 1 µ β(t 1)] + e t. (2.24) By simple algebra, the model (2.24) becomes Y t = [µ(1 ρ)+βρ]+β(1 ρ)t + ρy t 1 + e t γ 0 + γ 1 t + ρy t 1 + e t which is the same model as (2.23). We denote the pivotal statistic associated with ˆρ τ as τ τ. ˆρ τ and τ τ can be obtained if we replace (Y t Ȳ )with(y t â ˆbt) in (2.1) and (2.3), respectively, where â and ˆb are OLS estimates of a and b in the simple linear regression Y t = a + bt + e t. (2.25)

31 19 Leybourne et al (1998) found that τ τ gives a similar size distortion pattern to that of τ µ when Model I is used to generate the data. Our empirical sizes for τ τ in Table 2.2 are in close agreement with theirs. The magnitude of the spurious rejection problem is more severe for τ τ than for τ µ for the smallest λ. The problem evaporates more rapidly for τ τ than for τ µ as λ increases. Denote the linear trend adjusted versions of τ s and τ w as τ s,τ and τ w,τ respectively. τ s,τ and τ w,τ can be obtained from (2.4) and (2.5), respectively, by replacing (Y t Ȳ ) with the residual from (2.25). Empirical sizes for τ s,τ and τ w,τ are given in Table 2.2. In the simulation, the number of replications is 5,000 and the sample size per replication is n = 100. Critical values used are those for the linear trend removed case in Fuller (1996). As in the mean adjusted cases in section 2.2.1, the spurious rejection problem is less severe for τ s,τ and τ w,τ than for τ τ. Also τ s,τ and τ w,τ show similar patterns of empirical sizes to τ s and τ w, respectively. τ s,τ gives a slightly higher rejection rate than τ s. Like τ w, τ w,τ gives satisfactory performance and retains size close to the nominal 5% level. Figure 2.4 displays the empirical sizes for model I with m = 10 for the linear trend adjusted statistics Using lagged first differences To extend the results for the first order process to more general processes, we consider an augmented test based on the WS estimator since it shows better performance than the other estimators in the previous results. We introduce lagged first differences as in the usual Augmented Dickey-Fuller test based on the OLS estimator. Our test statistic is denoted as τ w,a. Using the data arrangement and weights in Table , Fuller (1996), with p =2, we perform the weighted regression estimation of the autoregressive model written as Y t = θ 1 Y t 1 + θ 2 Z t 1 + e t

32 20 where Z t 1 = Y t 1 Y t 2. For reference, we reproduce the table for p = 2 here. Weight Dependent Variable θ 1 θ 2 w 3 Y 3 Y 2 Y 2 Y 1 w 4 Y 4 Y 3 Y 3 Y w n Y n Y n 1 Y n 1 Y n 2 1-w n 1 Y n 2 Y n 1 Y n 1 Y n 1-w n 2 Y n 3 Y n 2 Y n 2 Y n w 2 Y 1 Y 2 Y 2 Y 3 The estimator of θ =(θ 1,θ 2 ) can be obtained as ˆθ =( θ 1, θ 2 ) =(X WX) 1 X WY where X is the (2n 2p) p matrix below the headings θ 1, θ 2 in the table, Y is the (2n 2p)-dimensional column vector called the dependent variable and W is the (2n 2p) diagonal matrix whose elements are given in the Weight column. The hypothesis of a unit root is tested by testing the hypothesis that θ 1 = 1. Then the test statistic τ w,a can be written as τ w,a = θ 1 1 s.e. (2.26) whose limiting distribution, using the proper s.e., is the same as that of τ w in (5.1). Because each observation gives rise to 2 rows in Fuller s Table, the standard error printed by a typical regression package needs to be multiplied by 2 in (2.26) in order to use the critical values from Fuller (1996). Empirical sizes for Model I as a DGP are shown in Table 2.3 and Figure 2.5. The same values of m and c as before are used. The number of replications is 1,000 and thesamplesizeisn= 100. As a whole, the performance of the WS estimator in the augmented case is as good as in the simple case. In particular, τ w,a maintains size close to the nominal 5% level regardless of the break size.

33 Data with a break in trend In this section, we consider another DGP Model II : Y t = Y t 1 + mσi(t >c)+e t, t =1,2,..., n where the e t are normal independent (0,σ 2 ) random variables and we can assume σ = 1 without loss of generality as in Model I. This model corresponds to Model (B) under the null hypothesis of Perron (1989). Since Model II can be rewritten as Y t = Y t 1 + e t, t =1,,c Y t = Y t 1 +m+e t = Y c +m(t c)+ t e j, t = c+1,,n, (2.27) j=c+1 it might be called a random walk with a break in drift where mi(t >c) is the drift parameter. When m 0, the time trend will dominate the long-run behavior of Y t in (2.27) because t j=c+1 e j = O p ( n) and hence t j=c+1 e j is small in probability relative to t. Therefore we consider the linear trend adjusted versions of statistics to test the unit root null hypothesis. Time series data generated from Model II with various values of m are displayed in Figure 2.6. Leybourne et al (1998) reported empirical sizes for Model II using the test statistic τ τ. Columns labelled τ τ in Table 2.4 contain our empirical sizes for τ τ,whichareclose to those of Leybourne et al (1998). They found that, as m increases, the range of values of λ for which serious problems are found increases. They also concluded that frequent spurious rejections of the null hypothesis occur for low values of λ but the pattern for Model II is a little different than that for Model I. The spurious rejection problem was most severe for the lowest values of λ in Model I. In contrast, for Model II, it increases at first in severity with increasing λ, but then rapidly diminishes as λ increases further. We use test statistics τ s,τ and τ w,τ to obtain the proportion of rejections of the unit root null hypothesis. We consider the same values of m and c as those used

34 22 by Leybourne et al (1998). These results are also presented in Table 2.4. We notice that, for Model II, τ s,τ and τ w,τ give similar patterns of empirical sizes to each other. Although they do not show size distortion problems, they are severely undersized for a wide range of c which might suggest lower power against the alternative hypothesis. We comment on this in section 2.4. As far as spurious rejection is concerned, τ w,τ again presents much better empirical sizes than τ τ. Empirical sizes for Model II with m = 2 are shown in Figure Empirical powers allowing for a break Thus far we have considered the effects of ignoring an existing break and have found the weighted symmetric estimator performs well, avoiding spurious declararations of stationarity. If a series consists of a level shift plus stationary errors and one tests for stationarity without modelling the level shift, it is unclear whether accepting or rejecting the null hypothesis is desirable - neither is correct so power should not be a concern here. However, if we modified the WS estimator to accomodate a model with a break, we would want that test to have good power. We investigate this now. Let D t be a dummy variable representing a structural break, that is D t = I(t >c). We first perform regression estimation of the model Y t = β 0 + β 1 D t + e t and produce the residuals r t = Y t Ŷt = Y t ˆβ 0 ˆβ 1 D t. Replacing (Y t Ȳ )with r t in (2.5) and (2.6), we obtain a WS estimator and associated pivotal statistic, respectively, allowing for a break. For the limiting distribution of these statistics, see Appendix A. Table 2.5 presents empirical size and power results with respect to Model I as a DGP. Various values of n, λ and ρ are used where λ = c/n is the break time in D t. Although we are assuming a known break, it makes no difference in the results

35 23 for the modified estimator whether there is a break or not in the DGP. Both critical values from a simulation method and from Fuller (1996) are applied. The number of replications is 1,000. For all values of n, empirical sizes (ρ = 1) are too big if we use the critical values from Fuller (1996). This is no surprise since most previous research on trend breaks shows that critical values must be adjusted for breaks. For instance, Perron (1989) showed that the critical values under various models are noticeably smaller than the standard Dickey-Fuller critical value. The values for τ w with no fitted break are shown under the label λ =0inTable 2.5. Previous researchers have used least squares estimation for trend break cases and, in Table 2.5, we include with underlines the powers obtained using OLS estimation. See Tables 4-6 in Pantula et al (1994) for the empirical powers of τ µ and τ w when no breakispresent. Using critical values from simulation, we obtain reasonably good empirical powers. For given n and ρ, asλchanges from.02 to.5, the power decreases. In general, when there really is no break, the modified test that is invariant to breaks gives less power than τ w with no fitted break. When there is a break (λ >0), the modified test based on the WS estimator gives better power than the modified test based on the OLS estimator. It is seen that substantial improvement in power occurs with the WS estimator especially for larger λ values. 2.5 Conclusion In this paper, we compare the empirical sizes of unit root tests based on the OLS estimator with those of tests based on alternative symmetric estimators. We find that the symmetric estimators have much less severe size distortion problems than the OLS estimator in both the random walk with a break in level (Model I) and the random walk with a break in trend (Model II).

36 24 The WS estimator in particular shows practically no size distortion and, even for large m, nearly retains the nominal level. The performance of the WS estimator in the augmented case is as good as in the simple case. The WS estimator also generates reasonably good empirical power when modified to allow for a break. Obviously we do not want to reject the unit root null hypothesis when it is true but there is a break. Therefore we recommend the use of tests based on the WS estimator to avoid the converse Perron phenomenon. Further, if the break is modeled, the WS estimator has good power properties.

37 25 Table 2.1: Empirical sizes for Model I using mean adjusted statistics m=2.5 m=5 m=10 c τ µ τ s τ w τ µ τ s τ w τ µ τ s τ w

38 26 Table 2.2: Empirical sizes for Model I using linear trend adjusted statistics m=2.5 m=5 m=10 c τ τ τ s,τ τ w,τ τ τ τ s,τ τ w,τ τ τ τ s,τ τ w,τ